Calculate the missing side of a right triangle using a² + b² = c².
The Pythagorean theorem is one of the most fundamental and well-known theorems in mathematics. It describes the relationship between the sides of a right triangle and has been used for over 2,500 years in construction, navigation, and countless other applications.
Where a and b are the legs (shorter sides) and c is the hypotenuse (longest side, opposite the right angle)
| Find | Formula | Example |
|---|---|---|
| Hypotenuse (c) | c = √(a² + b²) | a=3, b=4: c = √(9+16) = √25 = 5 |
| Side a | a = √(c² - b²) | b=4, c=5: a = √(25-16) = √9 = 3 |
| Side b | b = √(c² - a²) | a=3, c=5: b = √(25-9) = √16 = 4 |
Pythagorean triples are sets of three positive integers (a, b, c) that satisfy the Pythagorean theorem. The most famous is (3, 4, 5).
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 20 | 21 | 29 |
| a | b | c | Factor |
|---|---|---|---|
| 6 | 8 | 10 | ×2 |
| 9 | 12 | 15 | ×3 |
| 12 | 16 | 20 | ×4 |
| 15 | 20 | 25 | ×5 |
While named after the Greek mathematician Pythagoras (c. 570-495 BCE), this relationship was known to ancient Babylonians and Egyptians long before. The theorem has hundreds of different proofs, more than any other mathematical theorem.
Right triangle with legs a, b and hypotenuse c
To verify a right triangle, check if:
a² + b² = c²
Example: 3² + 4² = 9 + 16 = 25 = 5²